Sangwine, SJ (2014) On harmonic analysis of vectorvalued signals. arXiv.org.

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Abstract
A vectorvalued signal in N dimensions is a signal whose value at any time instant is an Ndimensional vector, that is, an element of $\mathbb{R}^N$. The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in Ndimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by Ndimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in $\mathbb{R}^N$. Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions.
Item Type:  Other 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science and Health > Computer Science and Electronic Engineering, School of 
Depositing User:  Jim Jamieson 
Date Deposited:  23 Jul 2015 15:55 
Last Modified:  17 Aug 2017 17:34 
URI:  http://repository.essex.ac.uk/id/eprint/14439 
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